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2(4x-12)²-7(4x-12)+3=0 equation
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The solution
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[src]
2 2*(4*x - 12) - 7*(4*x - 12) + 3 = 0
$$\left(2 \left(4 x - 12\right)^{2} - 7 \left(4 x - 12\right)\right) + 3 = 0$$
Detail solution
Expand the expression in the equation
$$\left(2 \left(4 x - 12\right)^{2} - 7 \left(4 x - 12\right)\right) + 3 = 0$$
We get the quadratic equation
$$32 x^{2} - 220 x + 375 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 32$$
$$b = -220$$
$$c = 375$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{15}{4}$$
$$x_{2} = \frac{25}{8}$$
$$\left(2 \left(4 x - 12\right)^{2} - 7 \left(4 x - 12\right)\right) + 3 = 0$$
We get the quadratic equation
$$32 x^{2} - 220 x + 375 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 32$$
$$b = -220$$
$$c = 375$$
, then
D = b^2 - 4 * a * c =
(-220)^2 - 4 * (32) * (375) = 400
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{15}{4}$$
$$x_{2} = \frac{25}{8}$$
Rapid solution
[src]
x1 = 25/8
$$x_{1} = \frac{25}{8}$$
x2 = 15/4
$$x_{2} = \frac{15}{4}$$
x2 = 15/4
Sum and product of roots
[src]
sum
25/8 + 15/4
$$\frac{25}{8} + \frac{15}{4}$$
=
55/8
$$\frac{55}{8}$$
product
25*15 ----- 8*4
$$\frac{15 \cdot 25}{4 \cdot 8}$$
=
375 --- 32
$$\frac{375}{32}$$
375/32